> I hope they have affordable accommodation.
I am afraid you are coming in to high season now, so nowhere is going to be cheap, particulalry the lake district which is one of the UKs most popular tourist destinations.
Maybe a YHA building would suit you. They have them all over the UK and are very cheap, some are very basic but others are more upmarket.
Some are shared accomodation (many beds in the same room) but in others you can get a private room.
Have a look at their web site here
http://www.yha.org.uk/